## Section: New Results

### Quasi-Riemannian approach to constrained optimization

Participants : Didier Bailly [Research Engineer, ONERA Department of Applied Aerodynamics, Meudon] , Jean-Antoine Désidéri.

In differentiable optimization, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is one of the most efficient methods for unconstrained problems. Besides function values, it only requires the specification of the gradient. An approximate Hessian is calculated by successive approximations as part of the iteration, using rank-1 correction matrices. As a result, the iteration has superlinear convergence : when minimizing a quadratic function in $n$ variables, if the one-dimensional minimizations in the calculated directions of search are done exactly, the Hessian matrix approximation is exact after $n$ iterations, and from this, the iteration identifies to Newton's iteration, and produces the exact local optimum in only one additional iteration ($n+1$ in total).

However the BFGS method does extend to constrained problems very simply.
Following Gabay [82] and other authors, Chunhong Qi *et al* [120]
have proposed a “Riemannian” variant, RBFGS
that indeed incorporates equality constraints in the formulation and actually demonstrates superior convergence rates for
problems with a large number of variables. However these Riemannian formulations are non trivial to implement since they require
procedures implementing non-trivial differential-geometry operators ('retraction' and 'metric transport') to be developed.
In their paper, they assume a formal expression of the constraint to be known.
But, in PDE-constrained optimization, many constraints are functional, and it is not clear how can the
metric transport operator in particular can be defined.

We are investigating how can a quasi-Riemannian method can be defined based on the sole definition of evaluation procedures for the gradients. By condensing all the equality constraints in one, a purely-explicit approximate retraction operator has been defined that yields a point whose distance to the contraint surface is fourth-order at least. The associated transport operator is currently being examined formally. These techniques will be experimented in the context of constrained optimum-shape design in aerodynamics [20]